Test of Causality in Conditional Variance Hafner and Herwatz Test for Causality

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Örebro University

School of Business and Economics

Statistics, Advanced level thesis, 15 hp

Supervisor: Professor Panagiotis Mantalos

Examiner: Professor Sune Karlsson

Spring 2015

Test for Causality in Conditional Variance

Hafner and Herwatz Test for Causality

Abdullah Jatta

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Abstract

In the past relations between asset returns of financial variables has been the nexus of researchers. However in recent years more attention and time has been dedicated to the study of relations in conditional variance. This is due to the fact that contagion effects are not only limited to returns but also to the conditional variance of asset returns. The aim of this thesis is to test for causality in conditional variance i.e. in the granger sense. The (Hafner & Herwartz, 2006) is used to test for causality and the conditional variance is modelled using DCC-GARCH (1, 1) of (Engle, 2002) . A simulation study with different sample sizes and different data generating processes is used for testing the size and power properties of the HH test. The test is then used on two different stock indices. The simulation study shows good size properties for all DGP’s. The test also shows good power properties overall. Regardless there are other short comings of the test which is explained in more detail in later chapters.

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Table of Contents

1. Introduction ...4 2. Literature review ...4 3. Theoretical basis ...5 3.1 ARCH ...5 3.2 GARCH ...5

3.3 VECTOR AUTOREGRESSIVE MODEL (VAR) ...6

3.4 MULTIVARIATE GARCH ...7

3.5 CONDITIONAL CORRELATION GARCH MODELS ...8

3.5.1 CCC-GARCH ...8

3.5.2 DCC-GARCH ...8

3.6 CAUSALITY TEST IN CONDITIONAL VARIANCE ...9

3.7 H.H test... 10

4 SIMULATION ... 11

4.1 Results from size simulation ... 13

4.2 Results from Power Simulation... 14

5 DATA DESCRIPTION... 17

5.1 Results of HH Test on Data ... 18

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1. Introduction

In recent times there is overwhelming evidence of economic integration between financial markets both nationally and internationally. Advancement in information and technology and the increase flow of capital is driving globalisation. A seemingly unrelated phenomenon in one market affects another in a different setting or geography. Having an idea and understanding of the relationships between these markets is becoming more and more important for investors and researchers. Often the relationship in returns is investigated. It is becoming more evident to as well focus on the volatility of asset returns. Volatility is crude way of measuring the standard deviation or variance of an underlying asset return (Brooks, 2008). The correct modelling of volatility improves the accuracy of parameter estimates and forecast. It is therefore important to correctly model and check the influence of second order moments in financial time series analysis. For two or more asset returns the effect of volatility spill over is of particular interest. Volatility spill over between markets is gauged using multivariate generalised autoregressive heteroskedastic (MGARCH) models. This brings us to the concept of causality in the granger sense (Granger, 1969). One variable (

r

1) does not granger cause another variable (

r

2) if past

shocks about (

r

1) have no influence in the forecast of (

r

2). Non granger causality in conditional

variance can therefore be defined as past volatilities of (

r

1) has no effect on the present volatility

of (

r

2) and vice versa.

2. Literature review

Volatility modelling has been the subject of numerous discussions since (Engle, 1982) introduced the autoregressive conditional heteroskedastic (ARCH) model. Numerous studies about ARCH have been performed with regards to univariate time series, for instance (Bollerslev, Chou, & Kroner, 1992), (Higgins, Bera, & others, 1992) and (Engle & Kroner, 1995). Globalisation has driven international markets closer and closer together to the extent of merging economies. This phenomena leads to new questions about volatility of markets. Financial institutions and academics alike begin to ponder whether volatility in one market spills to another market. Likewise does the volatility of one asset return has any effect on another asset return, for instance (Bauwens, Laurent, & Rombouts, 2006). Further questions such as if negative and positive shocks have the same impact on volatility and whether volatility is relayed through the conditional variance of asset returns.

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The multivariate GARCH models have been largely regarded as the blueprint to answering the questions raised in the subsequent chapter. For the purpose of understanding we explain models leading to the multivariate GARCH model used in this thesis which is the DCC-GARCH (1, 1)

3. Theoretical basis

3.1 ARCH

The ARCH model of (Engle, 1982) is the first model geared towards explaining volatility. It is built on the assumption that

(1) The shock at of an asset return is serially uncorrelated but dependent

(2) The dependence of at can be explained by a simple quadratic function of its lagged values i.e.

at = ht1/2εt, ht = A0 +A1a2t-1 + . . . + Ama2t-m where εt iid (0,1) and A0 > 0 ; Ai ≥ 0

NOTE: εt can be normal or t-distributed in practice

From the formulation of the ARCH model, large lagged shocks will generate large conditional variance ht for the shock at.

There are weaknesses in modelling volatility using ARCH. Some of the weaknesses include: (1) From the structure of the model, positive and negative shocks have the same impact on

volatility because of the square of past shocks.

(2) ARCH is restrictive. Example

A

12 of the ARCH(1) should be within the range [0, ], for

the series to have a finite fourth moment, (Tsay, 2010)

(3) Very likely to over predict volatility because they respond slowly to large isolated shocks to the return series, (Tsay, 2010)

3.2 GARCH

Due to the limitations of the ARCH model, (Bollerslev, 1986) proposed an extension to ARCH called the generalised autoregressive heteroskedatic (GARCH) model. The innovation of an asset return at for GARCH is modelled as

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at =ht1/2εt ; ht=A0+ ∑mi=1Aia2t-i + ∑sj=1 Bjht-j

εt iid (0, 1)

A0 > 0; Ai ≥ 0; ∑i=1(m,s) (Ai+Bj) < 1

The constraint on (Ai+Bj) shows that at has a finite unconditional variance while the conditional variance may blow up if the sum is equal or exceeds unity.

The equation ht reduces to ARCH (m) when s=0. That is to say when s=0, the extension

from ARCH to GARCH is broken. The parameters Ai and Bj are referred to as ARCH and

GARCH parameters respectively. The simple GARCH (1, 1) is of the form.

ht=A0 + A1a2t-1 + B1ht-1; 0 ≤ A1, B1 ≤ 1, (A1 + B1) < 1

There are different formulations of the GARCH model but for the purpose of this thesis we limit to the GARCH (1, 1).

3.3 VECTOR AUTOREGRESSIVE MODEL (VAR)

The VAR model is used to model the relationship between two time series. It facilitates the analysis of spill over between two markets. The simple VAR process of order 1 is of the form

rt= φ0+ Φ1rt-1 + at

Where φ0 is k dimensional vector of constants, Φ1 is a k x k matrix and at is a serially

uncorrelated random vector with mean zero and a positive definite covariance matrix Σ. The bivariate case with rt = (r1t, r2t) and at = (a1t, a2t) will consist of the following equations

r1t= φ10 + Φ11r1,t-1 + Φ12r2,t-1 + a1t

r2t= φ20 + Φ21r1,t-1 + Φ22r2,t-1 + a2t

Φ12 denotes the conditional effect of r2,t-1 on r1t given r1,t-1. That is to say Φ12 captures the

linear dependence of r1t on the lagged values of r2t given the effect of the lagged values of r1t

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Φ21 captures the conditional effect of r1,t-1 on r2 given r2,t-1. The same thinking as before, Φ21

captures the linear dependence of r2t on the lagged values of r1t given the effect of the lagged

values of r2t on r2t.

Φ11 denotes the effect of r1,t-1 on r1t and Φ22 denotes the effect of r2,t-1 on r2t.

3.4 MULTIVARIATE GARCH

In the VAR model we postulated the shock at has a mean zero and a positive definite covariance

matrix Σ. It is well known that volatility clustering occurs in asset returns i.e. the fluctuations in returns vary over time. It will therefore be useful to model the second order time varying

moments. Now if instead each of the shock in at is modelled by a GARCH (1, 1), we then have

a multivariate GARCH model. The GARCH (1, 1) is well known to capture the dynamics in second order moments. (Bollerslev et al., 1992) asserts that GARCH (1, 1) is sufficient for modelling the variance dynamics in long sample periods. (Engle & Kroner, 1995) also asserts that GARCH (1, 1) is the leading model for modelling returns.

Different multivariate GARCH models exist depending on the structure of the conditional variance ht. There are generalizations from univariate to multivariate cases, like the diagonal

VEC model of (Bollerslev, Engle, & Wooldridge, 1988) and the (BEKK,1990) model. The idea behind these models is the direct modelling of the covariance matrix. An alternative to the direct

modelling approach is the modelling of conditional variances and correlations. The Ht matrix is

decomposed as Ht=DtRtDt

Where Dt = diag (h1t1/2, h2t1/2, . . ., hNt1/2) is conditional standard deviation and Rt is a correlation

matrix.

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3.5 CONDITIONAL CORRELATION GARCH MODELS

3.5.1 CCC-GARCH

The first of this model is the constant conditional correlation GARCH (CCC-GARCH) of

(Bollerslev, 1990). In this model, the time varying component of Ht does not vary with time.

Rt = R

The shock at is modelled as a univariate GARCH.

at=ht1/2ϵt; ht = A0 + ∑i=1mAiat-i2 + ∑sj=1Bjht-j: ϵt is iid(0, R)

ht = [h1,t, . . ., hN,t]’, at2 = [ , . . ., ]’, A0 is an (N x 1) vector of constants

Ai and Bj is an (N x N) matrices with elements such that the elements of ht are positive.

3.5.2 DCC-GARCH

The dynamic conditional correlation (DCC) GARCH of ( Engle & Sheppard, 2001) is of the

same form as the CCC-GARCH. The difference is that both Dt and Rt are time variant. Let at be

the shocks from n asset returns with mean 0 and covariance matrix Ht. The DCC-GARCH will

have the following definitions.

rt= µt + at ; at = Ht1/2εt ; Ht=DtRtDt

Ht is an n x n conditional variance matrix of at at time t

Dt is an n x n diagonal matrix of standard deviations of at at time t εt is an iid vector of errors with mean 0 an variance 1.

Rt is an nxn conditional correlation matrix of at at time t

Dt = diag (h1t1/2, . . . , hnt1/2). Note that the elements of Dt are modelled as GARCH (1, 1) as:

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The conditional correlation matrix Rt is obtained from the standardized residuals of the error

term εt. It is a symmetric matrix. The restriction on the matrix Ht is that it should be positive

definite and this happens when Rt is positive definite. In addition all the elements of Rt must be

less than or equal to unity. To make sure the above requirements are fulfilled Rt is decomposed

as

Rt = Qt*-1QtQt*-1 ; Qt= (1 – α - β ) + αεt-1ε’t-1+ βQt-1

= E( εtεt’ ) is the unconditional covariance matrix of the standardised residuals Qt*= diag( 11, . . . , nn)

α and β are scalars that should be greater than zero and there sum less than 1 for Ht to be

positive definite i.e.

α > 0 ; β > 0 ; α + β < 1

NOTE: We should at this point NOTE that the asymptotic properties of normality for the DCC-GARCH have not been proven, (Caporin & McAleer, 2013).

3.6 CAUSALITY TEST IN CONDITIONAL VARIANCE

Granger causality test is often employed to assess the linear relationship between time series variables. The fact remains that volatility in financial data is time varying, hence it is plausible to model second order moments.

Different methodology to test no causality in variance exists, with the goal to test for interaction in conditional variance. (Cheung & Ng, 1996) used the cross correlation function of the squared univariate GARCH residual estimates. (Lundbergh & Teräsvirta, 2002) constructed a test on the ARCH in GARCH principle. (Nakatani & Teräsvirta, 2009) formulated a test based on

parameter restrictions of the GARCH parameters. The size and power properties of the (Hafner & Herwartz, 2006) test being evaluated for this thesis adopted a framework of LM

specification testing in univariate GARCH models introduced by (Lundbergh & Teräsvirta, 2002) to test for causality in variance.

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3.7 H.H test

(Hafner & Herwartz, 2006) use the idea of “ARCH in GARCH” test, to obtain a LaGrange multiplier test for no-causality in conditional variance. The error vector at = (ait , ajt) is modelled

as follows. For i,j = 1,…,N , i ≠ j

ait = εit(hitgt)1/2 ; gt = 1 + πz’jt ; zjt = (a2jt-1, hjt-1)’ ; εit is iid(0, 1)

Where hit =ωi0 + ηia2i,t-1 + γihi,t-1. From the above parameterization of at, a sufficient condition

for the null hypothesis of no causality is for π=0. The null and alternative hypothesis of the HH test is thus constructed as

H0: π = 0 Vs H1: π ≠ 1

The LM statistic can be constructed by means of estimated univariate GARCH process. The score of the Gaussian log-likelihood function of ait is given by Xit(ε2it-1)/2 , Xit = hit(

), θi = (ωi0,ηi, γi)’

The following test statistics is proposed

λLM = ( -1) z'jt) V(θi)-1 ( -1) zjt) χ2(2) where V(θi) = ( z ' jt – x’it ( x’it)-1 z’jt) , k = -1)’

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4 SIMULATION

A Monte Carlo simulation was carried out to test the size and power of the HH test. For the size calculations data was generated from the DCC-GARCH (1, 1) where the off diagonal elements of the matrices Ai and Bj are zero. This is known as the diagonal method. For the power

calculations data was generated from the DCC-GARCH (1, 1) with non-zero off diagonal elements, known as the extended method. The simulation process was carried out in such way as to evaluate the impact of different persistence on the HH test. Two different unconditional correlation matrices were also used to gauge its influence on the test. The data was generated from both the t-distribution and normal distribution to see if it had any impact on the test. Furthermore for the power calculations the values of the off diagonal elements were interchanged for the one directional spill over case. The simulation was implemented with the CCGARCH package in R. I should as well mention that the ARCH and GARCH matrices are similar to that of (Nakatani, 2010) with changes to suit this thesis. The following shows the DGP’s for both power and size calculations.

For the size calculations 3000 replicates with sample sizes of 1000, 3000 and 5000 was performed for three different data generating processes as follows. Two different unconditional correlations of 0.48 and 0.70 was used.

A0= (0.02, 0.02), a vector of constants

A and B are the ARCH and GARCH parameters respectively

(α, β)= (0.05, 0.93), a vector of initial values for the DCC parameters (2x1)

Unconditional correlation = 0.48 and 0.70

Low persistence

A=

B=

12 Moderate persistence A= B= High persistence A= B=

For the power calculations the unconditional correlation 0.48 and 0.70 is common throughout. For any given DGP the same values are used for the different direction of spill over except

interchanging the entries of A12 and A21 for the ARCH matrix and B12 and B21 for the GARCH.

This accounts for the direction of the spill over from

r

1 to

r

2 and vice versa. Sample sizes of

1000, 2000, 3000, 4000 and 5000 with 3000 replicates were carried out.

Spill over from

r

1 to

r

2

A0 = (0.01, 0.02), vector of constants

Initial values of DCC (α, β )= (0.05, 0.93)

DGP1:High persistence, small off diagonal elements

A=

B=

DGP3: High persistence, moderate diagonal entries

A=

, B=

DGP5 High persistence, b21 high A=

B=

DGP7: Low persistence, high off diagonal elements

A=

, B=

Spill over from

r

2 to

r

1

A0= (0.02, 0.01),

Initial values of DCC (α, β) = (0.05, 0.93)

DGP2: High persistence, small off diagonal elements

A=

B=

DGP4: High persistence, moderate diagonal entries

A=

, B=

DGP6: High persistence, b21 high A=

B=

DGP8: Low persistence, high off diagonal elements

A=

, B=

13 Two way spill over

A0= (0.01, 0.02), (α, β)= (0.05, 0.93)

High persistence, low off diagonal entries

A=

, B=

High persistence, high off diagonal entries

A=

, B=

4.1 Results from size simulation

Estimation results from Monte Carlo simulation shows good size properties for the HH test, table 4.1-4.3. All DGP’s attain the nominal significance level of 0.05. In fact small p values are attained as the sample size increases. This trend is seen throughout regardless of the type

of persistence, value of the unconditional correlation or whether the innovations are t-distributed or normally distributed.

Table 4.1: Estimated size off HH test in simulation with Low persistence

Obs Corr normal t-distributed

1000 0.489 0.044 0.037 3000 0.489 0.034 0.027 5000 0.489 0.042 0.028 1000 0.700 0.035 0.023 3000 0.700 0.023 0.019 5000 0.700 0.030 0.018

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Table 4.2: Estimated size of HH test in simulation with moderate

persistence

Obs Corr normal t-distributed 1000 0.489 0.051 0.038 3000 0.489 0.041 0.034 5000 0.489 0.044 0.035 1000 0.700 0.046 0.030 3000 0.700 0.032 0.027 5000 0.700 0.043 0.026

Table 4.3: Estimated size of HH test in simulation with high persistence

Obs Corr normal t-distributed 1000 0.489 0.064 0.074 3000 0.489 0.051 0.055 5000 0.489 0.048 0.053 1000 0.700 0.070 0.072 3000 0.700 0.056 0.050 5000 0.700 0.059 0.052

4.2 Results from Power Simulation

In evaluating the power of the test, let’s begin by assessing the power in the spill over case from

r

1 to

r

2. For a start DGP1 table 4.2.1, with low off diagonal entries show poor power

properties. This could be attributed to the sensitivity of the DCC-estimation to zero or near zero off diagonal entries. The evidence is seen in DGP2 table 4.2.3, as a small increase of 0.001 (DGP1) to 0.004 (DGP2) brings about a significant surge in the power of the test. The power of the test increases with increasing sample size but it should as well be noted that for small off diagonal entries there is a noticeable difference between t-distributed innovations and normally distributed innovations. This phenomenon tends to diminish when the off diagonal entries are high table 4.2.7. Overall the HH test has good power properties with regards to both t-distributed and normally distributed innovations. The value of the unconditional correlation equally does not have much bearing on the power of the test. Spill over from

r

2 to

r

1 carries most of the attributes to its opposite case. The main difference is

with regards to small off diagonal entries where spill over from

r

2 to

r

1 shows lesser power.

The power in the two way spill over is no exception. It too has the same attributes to the first two cases.

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Table 4.2.1: Estimated power of

the HH test with high persistence,

small off diagonal entries (spill over from r1 to r2)

Table 4.2.2: Estimated power of

the HH test with high persistence,

small off diagonal entries (spill over from r2 to r1)

Obs Corr Normal t-distr.

1000 0.48 0.076 0.074 2000 0.48 0.071 0.092 3000 0.48 0.084 0.114 4000 0.48 0.091 0.140 5000 0.48 0.089 0.151 1000 0.70 0.062 0.062 2000 0.70 0.075 0.075 3000 0.70 0.086 0.086 4000 0.70 0.107 0.107 5000 0.70 0.108 0.108

Obs corr Normal t-distr.

1000 0.48 0.061 0.050 2000 0.48 0.056 0.055 3000 0.48 0.056 0.054 4000 0.48 0.049 0.045 5000 0.48 0.043 0.052 1000 0.70 0.060 0.046 2000 0.70 0.054 0.049 3000 0.70 0.060 0.050 4000 0.70 0.051 0.046 5000 0.70 0.049 0.052

Table 4.2.3: Estimated power of HH test with high persistence, moderate diagonal entries (spill over: r1 to r2)

Table 4.2.4: Estimated power of HH test with high persistence, moderate diagonal entries (spill over: r2 to r1)

Obs Corr Normal t-distr.

1000 0.48 0.139 0.201 2000 0.48 0.217 0.362 3000 0.48 0.333 0.530 4000 0.48 0.410 0.672 5000 0.48 0.502 0.769 1000 0.70 0.109 0.135 2000 0.70 0.166 0.227 3000 0.70 0.241 0.353 4000 0.70 0.301 0.457 5000 0.70 0.367 0.574

Table 4.2.5: Estimated power of HH test with high persistence and GARCH parameter B21 high (spill over: r1 to r2)

Obs corr Normal t-distr.

1000 0.48 0.072 0.100 2000 0.48 0.116 0.141 3000 0.48 0.142 0.180 4000 0.48 0.169 0.231 5000 0.48 0.190 0.273 1000 0.70 0.070 0.074 2000 0.70 0.093 0.096 3000 0.70 0.119 0.118 4000 0.70 0.130 0.138 5000 0.70 0.145 0.158

Table 4.2.6: Estimated power of HH test with high persistence and GARCH

parameter B12 high (spill over: r2 to r1)

Obs Corr Normal t-distr.

1000 0.48 0.545 0.506 2000 0.48 0.890 0.732 3000 0.48 0.988 0.815 4000 0.48 0.999 0.866 5000 0.48 1.000 0.889 1000 0.70 0.301 0.224 2000 0.70 0.604 0.337 3000 0.70 0.842 0.433 4000 0.70 0.950 0.504 5000 0.70 0.986 0.566

Obs corr Normal t-distr.

1000 0.48 0.383 0.351 2000 0.48 0.703 0.443 3000 0.48 0.916 0.466 4000 0.48 0.980 0.483 5000 0.48 0.997 0.512 1000 0.70 0.201 0.159 2000 0.70 0.392 0.169 3000 0.70 0.629 0.181 4000 0.70 0.801 0.196 5000 0.70 0.905 0.21

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Table 4.2.7: Estimated power of HH test with low persistence and high off

diagonal entries (spill over: r1 to r2)

Table 4.2.8: Estimated power of HH test with low persistence and high off

diagonal entries (spill over: r2 to r1)

Obs Corr Normal t-distr.

1000 0.48 0.584 0.553 2000 0.48 0.947 0.857 3000 0.48 0.997 0.954 4000 0.48 1.000 0.983 5000 0.48 1.000 0.992 1000 0.70 0.358 0.257 2000 0.70 0.730 0.424 3000 0.70 0.925 0.571 4000 0.70 0.990 0.691 5000 0.70 0.999 0.792

Obs Corr Normal t-distr. 1000 0.48 0.284 0.331 2000 0.48 0.513 0.606 3000 0.48 0.736 0.799 4000 0.48 0.857 0.914 5000 0.48 0.944 0.964 1000 0.70 0.227 0.235 2000 0.70 0.402 0.456 3000 0.70 0.603 0.651 4000 0.70 0.745 0.785 5000 0.70 0.863 0.890

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DATA DESCRIPTION

The datasets being used are the stock market prices of S&P 500 and DAX. The datasets are obtained from yahoo finance for the sample period 01 February 1995 to 31 March 2015. They are daily data of five trading days. All non-trading days were deleted. There is no theoretical basis for deleting that I am aware of except for convenience in this thesis. The different trading hours and different time zones may have the potential to induce some spurious effects. The effect is however assumed to be minimal due to speed of information exchange. The total number of observations under study is 4993. The price series are converted to return series by

r

t = ( / −1 )

Table 5.1: Summary Statistics of

S&P 500 and DAX Stock Returns

Index S&P 500 DAX

Mean 0.00029190 0.00030190 Median 0.000690 0.000836 Minimum -0.09469512 -0.07335522 Maximum 0.10957200 0.10797470 Std Deviation 0.01224660 0.01492602 Skewness -0.24316720 -0.009132 Kurtosis 8.020691 4.002314 Jarque-Bera 13447.61 3337.263 p-value (0.000) (0.000) LB-Q(10) 63.3686 24.84713 p-value (0.000) (0.005) LB-Q(25) 118.7016 65.75685 p-value (0.000) (0.000)

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From table 5.1 above S&P 500 and the DAX have fairly the same mean return. Judging by the standard deviations of the two series the DAX is slightly more volatile than the S&P 500. Both series indicate negative skewness, implying more negative returns. The jarque-bera test for the two series indicates that we should reject the null hypothesis of normality at the 5% significance level.

JB = (s2 + )

S and K are for skewness and kurtosis respectively

The Ljung-Box test of autocorrelation is also rejected. The results show that there is serial correlation up to lag m.

QLB= N(N+2)∑mk=1(rk2/N-k)

Another statistics checked is whether the series are stationary or not.(Brooks, 2008) explained that non-stationary series leads to spurious regression. The augmented dickey fuller test was used to check for unit root in the series. The null hypothesis is that unit root exist against the alternative that there is no unit root.

Table 5.2: Results of the ADF Test on S&P 500 and DAX Stock Returns

S&P 500 DAX

ADF(Lag 17) -17.09 -16.554

p-value 0.01 0.01

From the table above we reject the null hypothesis of unit root. Both series are stationary.

5.1 Results of HH Test on Data

The application of the HH test to the stock return gives evidence of causality with small p-value, table 5.1.1 below. The null hypothesis is rejected at the 0.05 significance level. The null hypothesis of no causality is χ2 distributed with 4 degrees of freedom. The test result shows that DCC estimation will be good for estimating the spill over effect. Table 5.1.2 shows results of the DCC estimation. It is interesting to have a close look at value of the

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associated standard error does not show evidence of spill over between the two stock indices which is not in agreement with the results of the HH test. This could be attributed to the weakness of the DCC estimation method. It was mentioned earlier that the DCC estimation is sensitive to low off diagonal entries and performs poorly if the values are close to zero. Thus if the second order conditional correlation between the indices is near zero then the DCC estimation will be unable to capture the association. Volatility spill over might be very small but nonetheless present.

Table 5.1.1: Result of HH test on the stock returns

Test stat 27.63542

P-value 1.48E-05

Table 5.1.2: Results of DCC estimation of the stock returns

(S&P 500 and DAX)

6 Conclusion & Suggestions

The HH test show good size and power properties for the different scenarios investigated in this thesis. It can therefore be used to test for spill over effects of second order moments. The flip side is that the test is not as powerful when the off diagonal entries of the GARCH matrix are small. This can be attributed to the limitation of the DCC estimation. It therefore remains

Variable estimates std. error

a1 0.0000008 0.0000004 a2 0.0000015 0.0128380 A11 0.0927794 0.0083178 A21 0.0040001 0.0134029 A12 0.0010000 0.0082444 A22 0.0867736 0.0000005 B11 0.8970971 0.0210793 B21 0.0010000 0.0127567 B12 0.0039998 0.0215179 B22 0.9054547 0.0139279 α 0.0128859 0.0122870 β 0.9853019 0.0144104

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to been seen how well it performs under other models different from the DCC-GARCH. The good thing about the H.H test is that the statistics can be derived under the null hypothesis of no causality which avoids implementation of complex models under the alternative hypothesis. An added advantage is that estimation of univariate GARCH (1, 1) models will suffice to implement the test.

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